

If you’d rather keep Vikunja as the backend and web UI, it has a CalDAV endpoint, so you can point DAVx5 at it and use Tasks.org or jtx Board on Android for the reminders themselves. That gets you native Android notifications without email, and you keep the web interface for planning. Vikunja’s CalDAV side is less mature than its web UI, though, so test whether due dates and reminders come across the way you expect before moving everything over.
For notifications that aren’t tied to a task app (say, “remind me when X happens” from a script or Home Assistant), ntfy is the simplest self-hosted push to Android, as mentioned above.

Adding to why it has to be reachable: when a local user follows someone on another server, that server pushes new posts to your server’s inbox, so the remote side has to be able to resolve your domain and connect over HTTPS. A LAN-only box can’t receive that.
The least painful compromise is a cheap domain (or a subdomain you control) plus a tunnel or a small VPS acting as reverse proxy, so nothing on your home network is exposed directly. Then close signups so only your household can register. If Mastodon feels heavy for a handful of users, GoToSocial is much lighter on RAM for that use case, though you’d use a separate client app with it.
One thing to decide up front: the domain is baked into every account and can’t be changed later without starting over.